Inverse scattering theory for the discrete PT-symmetric nonlocal nonlinear Schr\"{o}inger equation under arbitrarily large nonzero boundary conditions

TL;DR

This paper develops an inverse scattering transform theory for a discrete PT-symmetric nonlocal nonlinear Schrödinger equation with large nonzero boundary conditions, revealing new soliton types and solution behaviors.

Contribution

It introduces a novel IST framework for large NZBCs in the discrete PT-symmetric nonlocal NLS, including new soliton solutions and detailed analytical properties.

Findings

Discovery of oscillating solitons not previously reported

Identification of breathers under large NZBCs

Multi-soliton collision dynamics involving oscillating dark/anti-dark solitons

Abstract

In this paper, the theory of inverse scattering transform (IST) is developed for the discrete PT-symmetric nonlocal nonlinear Schr\"{o}inger equation under large nonzero boundary conditions (NZBCs). By considering that the data at infinity have constant amplitudes, two cases are studied where the previous IST theory fails for large NZBCs. Based on a suitable uniformization variable, the rigorous proofs for the analyticity, symmetries and asymptotic behaviors of the eigenfunctions and scattering coefficients are provided for the direc problem, and the potential reconstruction formula is derived by solving the Riemann-Hilbert problem. Particularly, the focusing equation is found to admit two types of novel solitons under large NZBCs: oscillating soliton and breather, where the former has not been previously reported, while the latter does not occur under small NZBCs. In addition, the multi-soliton solutions are shown to exhibit the collisions among oscillating dark/anti-dark solitons, and the superposition of oscillating soliton and breather.